edoc: No conditions. Results ordered -Date Deposited. 2024-07-14T20:59:36ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2021-06-09T07:17:12Z2021-08-09T10:09:02Zhttps://edoc.unibas.ch/id/eprint/81400This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/814002021-06-09T07:17:12ZA Note on the Lagrangian Flow Associated to a Partially Regular Vector FieldGianluca CrippaSilvia Ligabue2021-04-08T19:51:53Z2023-01-20T08:07:00Zhttps://edoc.unibas.ch/id/eprint/82582This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/825822021-04-08T19:51:53ZA note on the Lagrangian flow associated to a partially regular vector fieldIn this paper we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form $$ b(t,x_1,x_2) = (b_1(t,x_1),b_2(t,x_1,x_2)) \in {\mathbb R}^{n_1}\times{\mathbb R}^{n_2} \,, \qquad (x_1,x_2)\in{\mathbb R}^{n_1}\times{\mathbb R}^{n_2}\,. $$ We assume that the first component $b_1$ does not depend on the second variable $x_2$, and has Sobolev $W^{1,p}$ regularity in the variable $x_1$, for some $p>1$. On the other hand, the second component $b_2$ has Sobolev $W^{1,p}$ regularity in the variable $x_2$, but only fractional Sobolev $W^{\alpha,1}$ regularity in the variable $x_1$, for some $\alpha>1/2$. These estimates imply well-posedness, compactness, and quantitative stability for the Lagrangian flow associated to such a vector field. Gianluca CrippaSilvia Ligabue2018-08-24T13:46:24Z2019-01-15T13:41:35Zhttps://edoc.unibas.ch/id/eprint/58742This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/587422018-08-24T13:46:24ZLagrangian solutions to the Vlasov-Poissosystem with a point chargeWe consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of [17], proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in [8] in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure. Gianluca CrippaSilvia LigabueChiara Saffirio